Algebra

# Algebraic Identities

The polynomial $9-4x^2-20xy-25y^2$ can be factorized as $(3+ax+by)(3+cx+dy),$ where $a,$ $b,$ $c$ and $d$ are real numbers. What is the value of $abcd$?

What is the value of $a+b+c$ if \begin{aligned} & x^2+8y^2-6xy-2yz+zx+4x-16y+4z \\ &= (x-ay+b)(x-cy+z)? \end{aligned}

If $a$, $b$ and $c$ are real numbers and $(x^2+2x)^2-79(x^2+2x)-80=(x+a)^2(x+b)(x+c)$ is an identity in $x$, what is the value of $a+b+c$?

What is the value of $a+b+c$ if \begin{aligned} & (x^2+3x)^2+10x^2+30x-56 \\ &= (x-1)(x+a)(x^2+bx+c)? \end{aligned}

What is the value of $a+b+c$ if \begin{aligned} & x^2+5xy+4y^2+5x+23y-6 \\ &= (x+ay-b)(x+y+c)? \end{aligned}

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